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Local Bernstein theory, and lower bounds for Lebesgue constants

Terence Tao

Abstract

Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions $\{ x+iy: x \in I, 0 \leq y \leq y_0 \}$, showing that Bernstein-type bounds with acceptable errors can continue to hold for functions holomorphic in such rectangles, bounded and real-valued on the lower edge of the rectangle, at most exponentially large on the upper edge, and at most double exponentially large on the vertical sides. As a consequence of these bounds, we are able to localize the Erdős lower bound $\sup_{x \in [-1,1]} λ(x) \geq \frac{2}π \log n - O(1)$ on the Lebesgue constant of interpolation on $C([-1,1])$ to shorter intervals $I$ than $[-1,1]$, answering a question of Erdős and Turán. By using suitably weighted versions of the residue theorem, we also obtain the asymptotically sharp lower bound $\int_I λ(x)\ dx \geq \frac{4|I|}{π^2} \log n - o(\log n)$ for integral variants of such constants, answering a further question of Erdős.

Local Bernstein theory, and lower bounds for Lebesgue constants

Abstract

Classical (or ``global'') Bernstein theory establishes sharp control on entire functions of exponential type that are bounded and real-valued on the real axis. We localize some of this theory to rectangular regions , showing that Bernstein-type bounds with acceptable errors can continue to hold for functions holomorphic in such rectangles, bounded and real-valued on the lower edge of the rectangle, at most exponentially large on the upper edge, and at most double exponentially large on the vertical sides. As a consequence of these bounds, we are able to localize the Erdős lower bound on the Lebesgue constant of interpolation on to shorter intervals than , answering a question of Erdős and Turán. By using suitably weighted versions of the residue theorem, we also obtain the asymptotically sharp lower bound for integral variants of such constants, answering a further question of Erdős.
Paper Structure (27 sections, 26 theorems, 280 equations, 12 figures)

This paper contains 27 sections, 26 theorems, 280 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Global Bernstein theory
  3. Local Bernstein theory
  4. Application to a problem of Erdős and Turán on Lagrange interpolation
  5. Main application, and a trigonometric toy model
  6. Proof sketch for the sup norm bound
  7. Proof sketch for the integral bound
  8. Acknowledgments and AI tool disclosure
  9. Notation
  10. Some Fourier, complex, and harmonic analysis
  11. The square wave
  12. The residue theorem
  13. Some inequalities on trigonometric polynomials
  14. Harmonic measure
  15. Proof of local Bernstein estimates
  16. ...and 12 more sections

Key Result

Theorem 1.4

Let $\lambda, A > 0$, and let $f \in {\mathcal{B}}^{\infty,\mathbb{R}}_\lambda$ with $\|f\|_{{\mathcal{B}}^{\infty,\mathbb{R}}_\lambda} \leq A$.

Figures (12)

  • Figure 1: The monic Chebyshev polynomial $P(x) = 2^{1-n} T_n(x)$ with $n=20$. Note the local sinusoidal behavior in the interior of the interval $[-1,1]$. Not displayed is the rapid (and non-sinusoidal) growth of $P$ outside of this interval; see \ref{['fig-pot']} for a depiction of that growth in (negative) log-scale. (Image generated by Gemini.)
  • Figure 2: The Lebesgue function $\lambda(x)$ for the polynomial in \ref{['fig-cheby']}, together with the predicted maximal value of $\frac{2}{\pi} \log n + C$, which holds up well in the bulk of $[-1,1]$ but becomes less accurate near the endpoints. The predicted mean of $\frac{4}{\pi^2} \log n + \frac{C'}{2}$, which is smaller by a factor of about $\frac{2}{\pi}$, is also shown. (Image generated by Gemini.)
  • Figure 3: The logical dependencies between the main results of this paper involving trigonometric polynomials (or functions of global exponential type). The spacing here is chosen to be consistent with that in \ref{['fig:flow']} below.
  • Figure 4: The logical dependencies between the main results of this paper involving polynomials (or functions of local exponential type). The results in boxes are analogous to the corresponding results in \ref{['fig:trig']}. Additional dependencies involving other propositions and lemmas are omitted to reduce clutter.
  • Figure 5: The square wave $\mathrm{sgn}(\cos x)$, together with the Fejér sum approximant \ref{['fourier-sum-fejer']} with $M=1, 2, 5,10, 20$. Note how the approximant, being the convolution of the square wave with a non-negative approximation to the identity (the Fejér kernel), stays bounded by $1$ in magnitude, avoiding the Gibbs phenomenon. (Image generated by Gemini.)
  • ...and 7 more figures

Theorems & Definitions (61)

  • Remark 1.1
  • Example 1.2: Sinusoids and sinc functions
  • Example 1.3: Trigonometric polynomial
  • Theorem 1.4: Global Bernstein theory
  • proof
  • Remark 1.5
  • Theorem 1.6: Local Bernstein theory
  • Remark 1.7
  • Remark 1.8
  • Example 1.9: Chebyshev polynomial
  • ...and 51 more